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Study on Rollover Index and Stability for a Triaxle Bus

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Abstract

Vehicle rollover, and its resulting fatalities, is an actively researched topic especially for multi-axle vehicles in the field of vehicle dynamics and control. This paper first presents a new rollover index for a triaxle bus to accurately evaluate its rollover possibility and then discusses the influence laws of the vehicle rollover dynamics to explore the mechanism of its stability. First, a six degree of freedom rollover model of the triaxle bus is developed, including lateral, yaw, roll motion of the sprung mass of the front/rear axle, and roll motion of the unsprung mass of the front/rear axle. Next, some key parameters of the vehicle rollover model are identified. A new rollover index is deduced according to the basics of vehicle dynamics, to predict vehicle rollover risk for the triaxle bus, which is verified by TruckSim. Furthermore, the influence laws of vehicle rollover dynamics by vehicle parameters and road parameters are discussed based on the simulation results. More importantly, the results show that the new method of modeling can precisely describe the rollover dynamics of the studied bus, and the proposed new index can effectively evaluate the rollover possibility. Therefore, this study provides a theoretical basis to improve anti-rollover ability for triaxle buses.

Introduction

Vehicle rollover is a serious safety accident. Among more than 6 million police-reported motor vehicle crashes that occurred in the United States in 2014, only 2.0% involved vehicle rollovers, with the proportion of rollover accidents leading to fatal crashes being 18.9% [1]. Some heavy vehicles are more likely to rollover due to their high center of gravity (CG), large mass and so on. Therefore, it is important for heavy vehicles to improve their active safety [2]. Triaxle buses, with a high center of gravity and long wheelbase, are often reported in rollover accidents with many passengers lost. As a result, accurately evaluating the possibility and exploring the mechanism of dynamic stability of a triaxle bus is important.

Over the years, it is necessary to establish an accurate and effective rollover dynamic model to predict vehicle roll motion. As a basic vehicle model, the two degrees of freedom (DOF) vehicle model with four wheels was used, which considered the road bank angles by Yu et al. [3, 4]. Also, the rollover dynamics state can be more accurately expressed by adding the roll plane model. So, a three-DOF model with lateral, yaw and roll motion is used by Zhang et al. [5]. Furthermore, the vertical motions of sprung and unsprung masses are added to the three-degree-of-freedom model by Jin et al. [6]. Considering the effect of the road excitation, a seven-DOF combined model was proposed based on a vehicle with a traditional chassis by Bao et al. [7]. In addition, a nonlinear eight-DOF model was put forward including longitudinal, lateral, yaw, roll, and four-wheel rotational motions [8, 9]. In order to reflect the vehicle dynamic characteristics in different directions, a dynamic model with sixteen-DOF was presented, including longitudinal, lateral, yaw, vertical, pitch, roll motions of the vehicle body; the rotational motion of four wheels, and vertical motions of the unsprung masses [10]. Na et al. proposed a new four-node membrane element model based on bending modification [11]. The verification and validation procedures are presented for the originally developed FE model of a Para transit bus, which was built by Bojanowski [12].

Although numerous research studies have been done on vehicle rollover, few have focused on triaxle buses. Wei Liu et al. [13] presented an integrated chassis stability control for the triaxle electric bus. In order to analyze the roll dynamics, an effective rollover index is indispensable. In recent years, many effective rollover indexes have been proposed. Vehicle speed is a key factor to prevent rollover, a safety speed model for rollover prevention is built under the condition of vehicle rapid rotation [14]. Lateral-load transfer ratio (LTR) is one of the most commonly used rollover indexes. To make the traditional LTR more accurate, the effect by uncertainties in the height of the vehicle’s center of gravity, road disturbances, and possible multiple sensor faults are considered based on a nonlinear two-wheel vehicular system [15]. Phanomchoeng et al. [16] developed a new rollover index using the lateral acceleration as well as the vertical acceleration to detect both tripped and untripped rollovers. For heavy articulated vehicles with multiple axles, Huang et al. [2] proposed a multiple-rollover-indices. Li et al. [17] introduced an improved predictive LTR (IPLTR) which was based on an 8-DOF nonlinear vehicle model. Furthermore, as a dominant factor in vehicle rollover accident, drivers’ operations should also be considered. An extended model of a driver-vehicle closed-loop system, with the time delay of driver’s operation was proposed by Jin et al. [18]. Li et al. [19] analyzed the vehicle lateral stability and anti-rollover, which considered the effect of road excitation. A rollover coefficient is defined by Odenthal et al. that basically depends on the lateral acceleration to activate the emergency steering and braking system [20]. Imine et al. [21] proposed an estimator to estimate the vehicle rollover stability, including roll angle, yaw rate, and lateral acceleration.

In addition, some actuation systems have been presented, such as active steering, differential braking, active suspension, active anti-roll bar, and integrated chassis control. Gaspar et al. [22] studied the brake systems to obtain the anti-roll moment to realize anti-rollover of heavy vehicles. The combination of steering and braking control is applied to avoid rollover by Odenthal et al. [20]. Vu et al. [23] used active anti-roll bar as an actuator to improve the roll stability of heavy vehicles. A novel control method of Invariance Control is applied by Wollherr et al. [24].

In this paper, taking a tour bus 3A as research object, a method of modeling a triaxle bus rollover dynamic is developed and, taking the deflection of the flexible frame into consideration, a new evaluation method of rollover is proposed to detect the possibility of the triaxle bus rollover. Also, the dynamic rollover stability of the triaxle bus is analyzed.

This paper is organized as follows: in Section 2, the dynamic model of triaxle bus rollover is established, including yaw, lateral, and roll motion. Then, the real axles of the triaxle bus are considered equivalent to a virtual rear axle to simplify the dynamics. Section 3 proposes a new rollover index is proposed to detect the rollover risk and it is also verified in two different cases. In addition, the roll dynamics stability of the triaxle bus is simulated in typical maneuvers, and the influence laws of rollover stability of vehicle parameters and driving factors are discussed in Section 4. Furthermore, the roll stability of anti-external disturbances is analyzed in Section 5. Finally, some conclusions are made in Section 6.

Roll Dynamic Model of the Triaxle Bus

Triaxle Bus Model

Given the exclusive features of a triaxle bus, such as the high center of gravity, the long wheelbase, the large number of passengers’ capacity, and the variable distribution of passengers having an impact on its roll property; the triaxle bus is divided into the front and the rear part. For a triaxle bus, the middle and rear axle are on the same side of the center of mass, and the distance between them is short such that the roll coupling can be neglected. Therefore, the middle and rear axle of the triaxle bus is equivalent to a virtual rear axle, as shown in Figure 1.

Figure 1
figure1

The equivalent model of the triaxle bus

The distance between the first axle and the virtual rear axle is the equivalent wheelbase, which will be obtained in Section 2.3. In addition, a twisted bar with a constant stiffness is assumed to link between the front axle and the virtual rear axle.

For the sake of simplicity, the influence of the pitching, the lateral wind, and longitudinal motion are neglected, the road profile is regarded as symmetric with respect to the x axle. Thus, a six-DOF vehicle model is established, as shown in Figure 2.

Figure 2
figure2

Dynamic model of the triaxle bus rollover

From D’Alembert’s principle, the motions of the above model are as follows.

Lateral motion:

$$ma_{\text{y}} - m_{\text{sf}} h_{\text{f}} \ddot{\varphi }_{\text{sf}} - m_{\text{sr}} h_{\text{r}} \ddot{\varphi }_{\text{sr}} = 2F_{{{\text{Y}}1}} \cos \delta + 2F_{\text{Yr}} .$$
(1)

Yaw motion:

$$I_{\text{Z}} \dot{r} = 2aF_{\text{Y1}} \cos \delta + M_{\text{r}} .$$
(2)

Roll motion of the front sprung mass of the front axle:

$$\begin{aligned} I_{\text{Xf}} \ddot{\varphi }_{\text{sf}} & = m_{\text{sf}} h_{\text{f}} a_{\text{y}} + m_{\text{sf}} gh_{\text{f}} \varphi_{\text{sf}} - k_{\text{f}} (\varphi_{\text{sf}} - \varphi_{\text{uf}} ) \\ & \quad - l_{\text{f}} (\dot{\varphi }_{\text{sf}} - \dot{\varphi }_{\text{uf}} ) + k_{\text{b}} (\varphi_{\text{sf}} - \varphi_{\text{sr}} ). \hfill \\ \end{aligned}$$
(3)

Roll motion of the sprung mass of the rear axle:

$$\begin{aligned} I_{\text{Xr}} \ddot{\varphi }_{\text{sr}} & = m_{\text{sr}} h_{\text{r}} a_{\text{y}} + m_{\text{sr}} gh_{\text{r}} \varphi_{\text{sr}} - k_{\text{r}} (\varphi_{\text{sr}} - \varphi_{\text{ur}} ) \nonumber \\ & \quad - l_{\text{r}} (\dot{\varphi }_{\text{sr}} - \dot{\varphi }_{\text{ur}} ) + k_{\text{b}} (\varphi_{\text{sr}} - \varphi_{\text{sf}} ). \hfill \\ \end{aligned}$$
(4)

Roll motion of the unsprung mass of the front axle:

$$\begin{aligned}& 2F_{\text{Y1}} h_{\text{c}} + m_{\text{uf}} \left( {h_{\text{uf}} - h_{\text{cf}} } \right)a_{\text{y}} = - m_{\text{uf}} g\left( {h_{\text{uf}} - h_{\text{cf}} } \right)\varphi_{\text{uf}} \hfill \\ &- k_{\text{f}} \left( {\varphi_{\text{sf}} - \varphi_{\text{uf}} } \right) - l_{\text{f}} \left( {\dot{\varphi }_{\text{sf}} - \dot{\varphi }_{\text{uf}} } \right){ + }k_{\text{uf}} \varphi_{\text{uf}} . \hfill \\ \end{aligned}$$
(5)

Roll motion of the unsprung mass of the virtual rear axle:

$$\begin{aligned}& 2(F_{\text{Y2}} + F_{\text{Y3}} )h_{\text{c}} + m_{\text{ur}} \left( {h_{\text{ur}} - h_{\text{cr}} } \right)a_{\text{y}} = k_{\text{ur}} \varphi_{\text{ur}} \hfill \\ &- m_{\text{ur}} g\left( {h_{\text{ur}} - h_{\text{cr}} } \right)\varphi_{\text{ur}} - k_{\text{r}} \left( {\varphi_{\text{sr}} - \varphi_{\text{ur}} } \right) - l_{\text{r}} \left( {\dot{\varphi }_{\text{sr}} - \dot{\varphi }_{\text{ur}} } \right), \hfill \\ \end{aligned}$$
(6)

where the lateral acceleration of the sprung mass is:

$$a_{\text{y}} = \dot{v} + ur,$$
(7)

and

$$\left\{ \begin{aligned} & F_{\text{Yr}} = F_{\text{Y2}} + F_{\text{Y3}} , \hfill \\ & M_{\text{r}} = 2bF_{\text{Y2}} { + }2cF_{{{\text{Y}}3}} . \hfill \\ \end{aligned} \right.$$
(8)

In above-mentioned equations, m denotes the total mass of triaxle bus. ms shows the total sprung mass. mf means the unsprung of mass vehicle. msf and msr represent the equivalent sprung mass of the front and the rear axle. muf and mur refer to the unsprung mass of the front and the rear axle. a, b and c are the longitudinal distance from the CG to the front axle, middle axle and rear axle, respectively. ay is the lateral acceleration. u is the longitudinal speed, and v denotes the lateral velocity. r denotes the yaw rate of the sprung mass. hf is the height between the center of front sprung mass and the roll center, while hr is the height between the center of rear sprung mass and the roll center. huf and hur are the height of the center of the front unsprung mass and the rear unsprung mass, measured upwards from the road. hcf and hcr are the height of the front roll center and the rear roll center, measured upwards from the road, respectively. IXf and IXr are the roll inertia of the front sprung mass and the rear sprung mass, measured about the roll axle. Iz is yaw inertia of the triaxle bus. \(\varphi_\text{sf}\) and \(\varphi_\text{sr}\) are the roll angle of the front sprung mass and the rear sprung mass. FYr the lateral force of the tires at the virtual axle, Mr is the yaw moment caused by the virtual rear axle. FY1, FY2 and FY3 are the lateral force of the tires at the first axle, the middle axle and the rear axle, respectively. kf and kr are the equivalent roll stiffness coefficient of the front suspension and the rear suspension. kuf and kur are the equivalent roll stiffness coefficient of the front unsprung mass and the rear unsprung mass. lf and lr are the equivalent roll damping coefficient of the front suspension and the rear suspension. kb is the torsion stiffness coefficient of vehicle frame.

In addition, the steering angle of front wheels δ is assumed to be sufficiently small that \(\cos \delta \approx 1\) in Eqs. (1) and (2) holds.

The lateral forces in Eqs. (1) and (2) mainly depending on the physical properties of the tire and the corresponding side slip angles βf, βm and βr observed on the front, middle and rear wheels. The slip angle can be determined from the simple geometric relations:

$$\left\{ \begin{aligned} & \beta_{\text{f}} = \arctan \left( {\frac{v + ar}{u}} \right) - \delta_{\text{f}} , \hfill \\ &\beta_{\text{m}} = { \arctan }\left( {\frac{v - br}{u}} \right), \hfill \\ &\beta_{\text{r}} = \arctan \left( {\frac{v - cr}{u}} \right). \hfill \\ \end{aligned} \right.$$
(9)

Simplify the tire model with the cornering stiffness of tires is constant, the lateral forces of tires can be obtained.

$$\left\{ \begin{aligned} & F_{\text{Y1}} = - K_{\text{f}} \beta_{\text{f}} , \hfill \\ & F_{{{\text{Y}}2}} = - K_{\text{m}} \beta_{\text{m}} , \hfill \\ & F_{{{\text{Y}}3}} = - K_{\text{r}} \beta_{\text{r}} , \\ \end{aligned} \right.$$
(10)

where Kf, Km and Kr are the cornering stiffness of the front wheels, the middle wheels and the rear wheels.

State Space Model of System

Setting \(U = \left[ {\begin{array}{*{20}c} {\varphi_{\text{sf}} } & {\varphi_{\text{sr}} } & {\varphi_{\text{uf}} } & {\varphi_{\text{ur}} } \\ \end{array} } \right]^{\text{T}} ,\) \(V = \left[ {\begin{array}{*{20}c} {\dot{\varphi }_{\text{sf}} } & {\dot{\varphi }_{\text{sr}} } \\ \end{array} } \right]^{\text{T}}\)and substituting Eqs. (9) and (10) into Eqs. (1)‒(8), the following equation of state space can be obtained:

$$M_{\text{q}} \left[ \begin{aligned} {\dot{v}} \hfill \\ {\dot{r}} \hfill \\ {\dot{U}} \hfill \\ {\dot{V}} \hfill \\ \end{aligned} \right] = A_{\text{q}} \left[ \begin{aligned} v \hfill \\ r \hfill \\ U \hfill \\ V \hfill \\ \end{aligned} \right] + B_{\text{q}} \delta_{\text{f}} ,$$
(11)

where

$$M_{\text{q}} = \left[ {\begin{array}{*{20}c} {M_{1} } & {0_{2 \times 4} } & {M_{2} } \\ {M_{3} } & {M_{4} } & {M_{5} } \\ {0_{2 \times 2} } & {M_{6} } & {0_{2 \times 2} } \\ \end{array} } \right],\quad A_{\text{q}} = \left[ {\begin{array}{*{20}c} {A_{1} } & {0_{2 \times 4} } & {0_{2 \times 2} } \\ {A_{2} } & {A_{3} } & {A_{4} } \\ {0_{2 \times 2} } & {0_{2 \times 4} } & {A_{5} } \\ \end{array} } \right],$$
$$B_{\text{q}} = \left[ {\begin{array}{*{20}l} {2K_{\text{f}} } \hfill & {2aK_{\text{f}} } \hfill & 0 \hfill & 0 \hfill & {2h_{\text{f}} K_{\text{f}} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]^{\text{T}} ,$$
$$M_{1} = \left[ {\begin{array}{*{20}c} m & 0 \\ 0 & {I_{\text{Z}} } \\ \end{array} } \right],\quad M_{2} = \left[ {\begin{array}{*{20}c} { - h_{\text{f}} m_{\text{sf}} } & { - h_{\text{r}} m_{\text{sr}} } \\ 0 & 0 \\ \end{array} } \right],$$
$$M_{ 3} = \left[ {\begin{array}{*{20}c} {h_{\text{f}} m_{\text{sf}} } & 0 \\ {h_{\text{r}} m_{\text{sr}} } & 0 \\ { - m_{\text{uf}} (h_{\text{uf}} - h_{\text{cf}} )} & 0 \\ { - m_{\text{ur}} (h_{\text{ur}} - h_{\text{cr}} )} & 0 \\ \end{array} } \right],\quad M_{4} = \left[ {\begin{array}{*{20}c} 0 & 0& 0& 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & {l_{\text{f}} } & 0 \\ 0 & 0 & 0 & {l_{\text{r}} } \\ \end{array} } \right],$$
$$M_{5} = \left[ {\begin{array}{*{20}c} { - I_{\text{Xf}} } & 0 \\ 0 & { - I_{\text{Xr}} } \\ 0& 0 \\ 0 & 0\\ \end{array} } \right],\quad M_{6} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0\\ 0 & 1& 0& 0 \\ \end{array} } \right],$$
$$A_{1} = \left[ {\begin{array}{*{20}c} {\frac{{ - \left( {2K_{\text{f}} + 2K_{\text{m}} + 2K_{\text{r}} } \right)}}{u}} & {\frac{{ - \left( {mu^{2} + 2aK_{\text{f}} - 2bK_{\text{m}} - 2cK_{\text{r}} } \right)}}{u}} \\ {\frac{{ - \left( {2aK_{\text{f}} - 2bK_{\text{m}} - 2cK_{\text{r}} } \right)}}{u}} & {\frac{{ - \left( {2a^{2} K_{\text{f}} + 2b^{2} K_{\text{m}} + 2c^{2} K_{\text{r}} } \right)}}{u}} \\ \end{array} } \right],$$
$$A_{2} = \left[ {\begin{array}{*{20}c} 0 & { - h_{\text{f}} m_{\text{s}} u} \\ 0 & { - h_{\text{r}} m_{\text{sr}} u} \\ {\frac{{ - 2K_{\text{f}} h_{\text{c}} }}{u}} & {\frac{{m_{\text{uf}} u^{2} \left( {h_{\text{uf}} - h_{\text{cf}} } \right) - 2aK_{\text{f}} h_{\text{cf}} }}{u}} \\ {\frac{{ - 2(K_{\text{m}} + K_{\text{r}} )h_{\text{c}} }}{u}} & {\frac{{m_{\text{ur}} u^{2} \left( {h_{\text{ur}} - h_{\text{cr}} } \right) + 2bK_{\text{m}} h_{\text{cr}} + 2cK_{\text{r}} h_{\text{cr}} }}{u}} \\ \end{array} } \right],$$
$$A_{3} = \left[ {\begin{array}{*{20}c} {k_{\text{b}} + k_{\text{f}} - m_{\text{sf}} gh_{\text{f}} } & { - k_{\text{b}} } & { - k_{\text{f}} } & 0 \\ { - k_{\text{b}} } & {k_{\text{b}} + k_{\text{r}} - m_{\text{sr}} gh_{\text{r}} } & 0 & { - k_{\text{r}} } \\ {k_{\text{f}} } & 0& { - k_{\text{uf}} + m_{\text{uf}} g(h_{\text{uf}} - h_{\text{cf}} ) - k_{\text{f}} } & 0 \\ 0 & {k_{\text{r}} } & 0 & { - k_{\text{ur}} + m_{\text{ur}} g(h_{\text{ur}} - h_{\text{cr}} ) - k_{\text{r}} } \\ \end{array} } \right]$$
$$A_{4} = \left[ {\begin{array}{*{20}c} {l_{\text{f}} } & 0\\ { 0_{{}} } & {l_{\text{r}} } \\ {l_{\text{f}} } & 0 \\ 0 & {l_{\text{r}} } \\ \end{array} } \right],\quad A_{5} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right].$$

Setting the state vector as \(x = \left[ {\begin{array}{*{20}c} {\dot{v}} & {\dot{r}} & {\dot{U}} & {\dot{V}} \\ \end{array} } \right]^{\text{T}}\). Then Eq. (11) can be rewritten into Eq. (12):

$$\dot{x} = Ax + B\delta_{\text{f}} ,$$
(12)

where \(A = M_{q}^{ - 1} \times A_{q} ,B = M_{q}^{ - 1} \times B_{q} .\)

Determination of the Equivalent Wheelbase

According to Refs. [25, 26], the method to determine the equivalent wheelbase is obtained. First, the linear two-DOF model of the vehicle can be set up based on Eqs. (1) and (2),

$$\left\{ {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}c} {\dot{\beta }} \\ {\dot{r}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\frac{{ - K_{{\text{f}}} - K_{{\text{m}}} - K_{{\text{r}}} }}{{mu}}} & {\frac{{ - aK_{{\text{f}}} + bK_{{\text{m}}} + cK_{{\text{r}}} }}{{mu^{2} }} - 1} \\ {\frac{{ - aK_\text{f} + bK_\text{m} + cK_\text{r} )}}{{I_{{\text{Z}}} }}} & {\frac{{ - a^{2} K_{{\text{f}}} - b^{2} K_{{\text{m}}} - c^{2} K_{{\text{r}}} }}{{I_{{\text{Z}}} u}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} \beta \\ r \\ \end{array} } \right]} \\ { \qquad \quad + \left[ {\begin{array}{*{20}c} {\frac{{K_{{\text{f}}} }}{{mu}}} \\ {\frac{{aK_{{\text{f}}} }}{{I_{{\text{Z}}} }}} \\ \end{array} } \right]\left[ {\delta _{{\text{f}}} } \right],} \\ {\beta = \frac{v}{u},} \\ \end{array} } \right.$$
(13)

where β represents side slip angle of mass center.

At steady state, \(\dot{\beta } = 0\) and \(\dot{r} = 0\), so

$$\begin{aligned} & \left[ {\begin{array}{l} \beta \\ r \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\frac{{ - K_{\text{f}} - K_{\text{m}} - K_{\text{r}} }}{mu}} & {\frac{{ - aK_{\text{f}} + bK_{\text{m}} + cK_{\text{r}} }}{{mu^{2} }} - 1} \\ {\frac{{ - aK_{\text{f}} + bK_{\text{m}} + cK_{\text{r}} }}{{I_{\text{Z}} }}} & {\frac{{ - a^{2} K_{\text{f}} - b^{2} K_{\text{m}} - c^{2} K_{\text{r}} }}{{I_{Z} u}}} \\ \end{array} } \right]^{ - 1} \hfill \\ & \times \left[ {\begin{array}{*{20}l} {\frac{{K_{\text{f}} }}{mu}} \\ {\frac{{aK_{\text{f}} }}{{I_{\text{Z}} }}} \\ \end{array} } \right]\left[ {\delta_{\text{f}} } \right]. \hfill \\ \end{aligned}$$
(14)

When only taking the front axle steering into account, the yaw rate gain can be described as follows:

$$\begin{aligned} \frac{r}{{\delta_{\text{f}} }}& = {{u\left[ {K_{\text{f}} K_{\text{m}} \left( {a + b} \right) - K_{\text{f}} K_{\text{r}} \left( {a + c} \right)} \right]} \mathord{\left/ {\vphantom {{u\left[ {K_{\text{f}} K_{\text{m}} \left( {a + b} \right) - K_{\text{f}} K_{\text{r}} \left( {a + c} \right)} \right]} {\{ K_{\text{f}} K_{\text{m}} \left( {a + b} \right)^{2} }}} \right. \kern-0pt} {\{ K_{\text{f}} K_{\text{m}} \left( {a + b} \right)^{2} }} \hfill \\ &\quad + K_{\text{f}} K_{\text{r}} \left( {a + c} \right)^{2} + K_{\text{m}} K_{\text{r}} \left( {c - b} \right)^{2} \hfill \\ &\quad - mu^{2} \left( {aK_{\text{f}} - bK_{\text{m}} - cK_{\text{r}} } \right)\} . \hfill \\ \end{aligned}$$
(15)

Setting

$$\left\{ \begin{aligned} l = a + c, \hfill \\ t = c - b. \hfill \\ \end{aligned} \right.$$
(16)

The yaw rate gain can be simplified.

$$\begin{aligned} &\frac{r}{{\delta_{\text{f}} }} = {u \mathord{\left/ {\vphantom {u {\left\{ {\left[ {K_{\text{f}} K_{\text{m}} \left( {l - t} \right)^{2} + K_{\text{f}} K_{\text{r}} l^{2} + K_{\text{m}} K_{\text{r}} t^{2} } \right]/} \right.}}} \right. \kern-0pt} {\left\{ {\left[ {K_{\text{f}} K_{\text{m}} \left( {l - t} \right)^{2} + K_{\text{f}} K_{\text{r}} l^{2} + K_{\text{m}} K_{\text{r}} t^{2} } \right]/} \right.}} \hfill \\ &\left[ {K_{\text{f}} K_{\text{m}} \left( {l - t} \right) + K_{\text{f}} K_{\text{r}} l} \right] + u^{2} \left[ { - m\left( {aK_{\text{f}} - bK_{\text{m}} - cK_{\text{r}} } \right)} \right]/ \hfill \\ &\left. {\left[ {K_{\text{f}} K_{\text{m}} \left( {l - t} \right) + K_{\text{f}} K_{\text{r}} l} \right]} \right\}. \hfill \\ \end{aligned}$$
(17)

So, the equivalent wheelbase of the triaxle bus can be obtained according to the equivalence of physical meaning as follows:

$$l_{\text{e}} = \frac{{K_{\text{f}} K_{\text{m}} \left( {l - t} \right)^{2} + K_{\text{f}} K_{\text{r}} l^{2} + K_{\text{m}} K_{\text{r}} t^{2} }}{{K_{\text{f}} K_{\text{m}} \left( {l - t} \right) + K_{\text{f}} K_{\text{r}} l}}.$$
(18)

Then, the equivalent front sprung mass is

$$m_{\text{sf}} = \frac{{m_{\text{s}} (l_{\text{e}} - a)}}{{l_{\text{e}} }}$$
(19)

and the equivalent rear sprung mass is

$$m_{\text{sr}} = \frac{{m_{\text{s}} a}}{{l_{\text{e}} }}.$$
(20)

Also, the equivalent front and rear unsprung mass can be described as

$$\left\{ \begin{aligned} & m_{\text{uf}} = \frac{{m_{\text{u}} (l_{\text{e}} - a)}}{{l_{\text{e}} }}, \hfill \\ & m_{\text{ur}} = \frac{{m_{\text{s}} a}}{{l_{\text{e}} }}. \hfill \\ \end{aligned} \right.$$
(21)

Parameter Estimation

The torsion stiffness coefficient kb is an important parameter of the vehicle model, which can be estimated based on steady-state value obtained from the simulation.

According to the above-mentioned roll motions of sprung mass (3), the torsion stiffness coefficient kb can be represented as follows:

$$\begin{aligned} k_{\text{b}} &= \{ [m_{\text{sf}} h_{\text{f}} a_{\text{y}} + m_{\text{sf}} gh_{\text{f}} \varphi_{\text{sf}} - k_{\text{f}} (\varphi_{\text{sf}} - \varphi_{\text{uf}} ) \hfill \\ & \quad - l_{\text{f}} (\dot{\varphi }_{\text{sf}} - \dot{\varphi }_{\text{uf}} ) - I_{\text{xf}} \ddot{\varphi }_{\text{sf}} - l_{\text{b}} (\dot{\varphi }_{\text{sf}} - \dot{\varphi }_{\text{sr}} )]\} / \hfill \\ & \quad (\varphi_{\text{sf}} - \varphi_{\text{sr}} ). \hfill \\ \end{aligned}$$
(22)

In a steady state response of the vehicle motion equation, the roll velocity and the roll acceleration of the sprung mass and unsprung mass are set to be zero. That is

$$\left\{ \begin{aligned} \dot{\varphi }_{\text{sf}} = 0, \hfill \\ \dot{\varphi }_{\text{uf}} =0, \hfill \\ \dot{\varphi }_{\text{sr}} =0, \hfill \\ \ddot{\varphi }_{\text{sf}} =0. \hfill \\ \end{aligned} \right.$$
(23)

The \(\varphi_\text{sf}\), \(\varphi_\text{sr}\), \(\varphi_\text{uf}\), ay can be obtained from TruckSim. Substituting these into Eq. (22):

$$k_{\text{b}} = \frac{{m_{\text{sf}} h_{\text{f}} a_{\text{y}} + m_{\text{sf}} gh_{\text{f}} \varphi_{\text{sf}} - k_{\text{f}} (\varphi_{\text{sf}} - \varphi_{\text{uf}} )}}{{\varphi_{\text{sf}} - \varphi_{\text{sr}} }}.$$
(24)

To demonstrate the accuracy of the above estimations, the torsion stiffness coefficient kb is calculated according to Eq. (4) in the same method. That is, the torsion stiffness coefficient also can be obtained as follows:

$$k_{\text{b}} = \frac{{m_{\text{sr}} h_{\text{r}} a_{\text{y}} + m_{\text{sr}} gh_{\text{r}} \varphi_{\text{sr}} - k_{\text{r}} (\varphi_{\text{sr}} - \varphi_{\text{ur}} )}}{{\varphi_{\text{sf}} - \varphi_{\text{sr}} }}.$$
(25)

From Eq. (24), the torsion stiffness coefficient can be estimated to 3967329 N·m/rad, which is close to the value calculated from Eq. (25). The error is less than 2%.

Validation of Vehicle Model

To verify the vehicle model, the dynamic performances of a three-axle bus rollover are simulated in TruckSim. A “tour bus 3A” model is used. The three-axle bus runs with the initial vehicle speed of 60 km/h, the front wheel steering angle at 6° in J-turn condition. The triaxle bus parameters are showed in Table 1.

Table 1 Parameters of a triaxle bus

Figure 3 shows the dynamic performances of the triaxle bus rollover. The solid lines are the results simulated by TruckSim model, and the dotted lines are calculated by the theoretical model.

Figure 3
figure3

Dynamic performances of a three-axle bus rollover: (a) Lateral acceleration, (b) Yaw rate, (c) Roll angle of front sprung mass, (d) Roll angle of rear sprung mass

As shown in Figure 3, the simulation and theoretical results are highly consistent with each other. In Figure 3(a), the peak value of lateral acceleration is 0.4613g simulated by the TruckSim model, and 0.4525g calculated by the theoretical model. Thus, the error of the peak value of lateral acceleration is 1.9%. Also, the steady state value and the reaction time of lateral acceleration can be compared, with errors of 3.8% and 2.2%, respectively.

Similarly, the error of the peak value, the steady state value and the reaction time of yaw rate can be obtained as 4.8%, 2% and 3.1% in Figure 3(b). The error of the peak value, the steady state value and the reaction time of roll angle of front sprung mass can be obtained as 3.4%, 2.3% and 5.5% in Figure 3(c). In addition, the error of the peak value, the steady state value and the reaction time of roll angle of rear sprung mass can be obtained as 2.7%, 2.9% and 5.7% in Figure 3(d).

From the above comparison, the error of vehicle dynamic characteristics between the theoretical model and the TruckSim model are less than 6%. As a result, the accuracy of the parameter estimation and the validity of the model has been verified.

Rollover Index of the Triaxle Bus

To prevent rollover of the triaxle, a precise and real-time rollover index is very necessary. The existing studies on the vehicle rollover mainly focused on the static stability factor (SSF). This measure of rollover propensity only reflects the most fundamental relation in a static condition and does not take the suspension and tire into account [27]. To detect rollover in the process of vehicle movement, the lateral-load transfer ratio (LTR) was commonly used by researchers [4,5,6,7].

The LTR was described as follows:

$$LTR = \frac{{F_{\text{L}} - F_{\text{R}} }}{{F_{\text{L}} + F_{\text{R}} }},$$
(26)

where, FL and FR are the vertical force of the left and right wheels. When the absolute value of LTR is equal to 1, the car is considered to roll over [28].

Establishment of Rollover Index

For the triaxle bus, because of its long wheelbase, the vertical load of the right and left wheels cannot be directly used to judge the rollover. Therefore, it is important to propose a rollover index for both the front and rear. According to the definition, the rollover index can be expressed as follows:

$$\left\{ \begin{aligned} & RI_{\text{f}} = \frac{{F_{L1} - F_{R1} }}{{F_{L1} + F_{R1} }}, \hfill \\ & RI_{\text{r}} = \frac{{F_{L2} + F_{L3} - (F_{R2} + F_{R3} )}}{{F_{L2} + F_{L3} + F_{R2} + F_{R3} }}, \hfill \\ \end{aligned} \right.$$
(27)

where RIf is the rollover index of front sprung mass system and RIr is the rollover index of rear sprung mass system.

Because the vertical forces of the wheels cannot be measured directly, the rollover index needs to be reformulated according to the developed vehicle model. The vertical acceleration of the tire is ignored. Then Eq. (28) is obtained:

$$\left\{ \begin{aligned} & F_{L1} - F_{R1} = F_{2} - F_{1} , \hfill \\ & F_{L1} + F_{R1} = m_{\text{f}} g, \hfill \\ \end{aligned} \right.$$
(28)

where F1 and F2 are the left and right suspension force. According to moment equilibrium equations, the difference value between F1 and F2 can be obtained.

$$\frac{{T_{\text{wf}} }}{2}\left( {F_{2} - F_{1} } \right) = - k_{\text{f}} \left( {\varphi_{\text{sf}} - \varphi_{\text{uf}} } \right) - l_{\text{f}} \left( {\dot{\varphi }_{\text{sf}} - \dot{\varphi }_{\text{uf}} } \right).$$
(29)

Combining Eq. (3),

$$\begin{aligned} F_{2} - F_{1} = \frac{2}{{T_{\text{wf}} }} \bigg[I_{\text{Xf}} \ddot{\varphi }_{\text{sf}} - m_{\text{sf}} h_{\text{f}} a_{\text{y}} \hfill \\ - m_{\text{sf}} gh_{\text{f}} \varphi_{\text{sf}} + k_{\text{b}} \left( {\varphi_{\text{sf}} - \varphi_{\text{sr}} } \right) \bigg]. \hfill \\ \end{aligned}$$
(30)

Thus,

$$\begin{aligned} RI_{\text{f}} & = - \frac{2}{{T_{\text{wf}} }} \bigg[I_{\text{Xf}} \ddot{\varphi }_{\text{sf}} - m_{\text{sf}} h_{\text{f}} a_{\text{y}} - m_{\text{sf}} h_{\text{f}} g\varphi_{\text{sf}} \hfill \\ & \quad + k_{\text{b}} (\varphi_{\text{sr}} - \varphi_{\text{sf}}) \bigg]/m_{\text{f}} g. \hfill \\ \end{aligned}$$
(31)

The RIr can be obtained in the same way,

$$\begin{aligned} & RI_{\text{r}} = - \frac{2}{{T_{\text{wr}} }} \bigg[I_{\text{Xr}} \ddot{\varphi }_{\text{sr}} - m_{\text{sr}} h_{\text{r}} a_{\text{y}} - m_{\text{sr}} h_{\text{r}} g\varphi_{\text{sr}} \hfill \\ &- k_{\text{b}} (\varphi_{\text{sr}} - \varphi_{\text{sf}})\bigg]/m_{\text{r}} g, \hfill \\ \end{aligned}$$
(32)

where Twf and Twr are the front wheel track and the rear wheel track.

RIf and RIr are not identical such that it is necessary to use the larger rollover index to indicate the rollover stability of the vehicle. A new rollover index of the triaxle bus is defined as

$$RI_{\text{t}} = \text{max} [\left| {RI_{\text{f}} } \right|,\left| {RI_{\text{r}} } \right|].$$
(33)

Basically, the triaxle bus is stable when the value of RIt is less than 1, and it is likely to rollover when the value of RIt is over than or equal to 1.

Validation of Rollover Index

To validate the proposed rollover index, the dynamic performance of the triaxle bus is simulated in TruckSim. The comparison of the LTR and RIt in two different cases is obtained in this section.

Case I. J-turn condition is used in this case. The front wheel steering angle is 6°. Therefore, the rollover indices of triaxle bus in J-turn can be obtained.

Figure 4(a) and (b) show comparisons of the rollover index of the front and rear sprung mass when the vehicle speed is 60 km/h. The dotted lines are obtained from the fundamental definition of LTR, and the solid lines are the new rollover indices, which are calculated by the above Eqs. (31) and (32). As shown in Figure 4(a), the maximum error between fundamental LTRf and RIf is less than 5%, and the peak value of RIf is a little bigger than LTR. That means RIf can be close to the traditional LTRf. In Figure 4(b), a good fit between the LTRr and RIr can be witnessed. In summary, the proposed rollover indices of front sprung mass and the rear sprung mass are efficient.

Figure 4
figure4

Comparison of new rollover index of front and rear sprung mass and LTR in J-turn maneuver: (a) Comparison of RIf and LTRf in J-turn; (b) Comparison of RIr and LTRr in J-turn

To better validate the effectiveness of RIt, the larger value of LTR of front and rear sprung mass is used to compare with RIt. As shown in Eq. (34):

$$LTR_{\text{t}} = \text{max} (\left| {LTR_{\text{f}} } \right|,\left| {LTR_{\text{r}} } \right|).$$
(34)

Figure 5 shows the roll stability of the triaxle bus when the vehicle speed is 90 km/h. The absolute value of both LTRt and RIt reach 1 at 2.1 s, and less than 1 a 3.5 s. That is, one side of the wheels lift off the ground from 2.1 s to 3.5 s. This further proves the accuracy of RIt.

Figure 5
figure5

Rollover dynamics of the triaxle bus in J-turn maneuver

Case II. In this case, another standard rollover test condition is implemented, Fishhook, with the amplitude of the front wheel steering angle set to 6°.

As shown in Figure 6, when the vehicle speed is 60 km/h, the theoretical rollover index of front sprung mass system and rear sprung mass make an agreement with the LTRf and the LTRr simulated by TruckSim. It demonstrates the validation of the proposed index in Fishhook condition.

Figure 6
figure6

Comparison of new rollover index of front and rear sprung mass and LTR in Fishhook maneuver: (a) Comparison of RIf and LTRf in Fishhook; (b) Comparison of RIr and LTRr in Fishhook

Figure 7 shows the roll dynamics of the triaxle bus in the Fishhook condition when the vehicle operates at 90 km/h. The RIt also agrees with the LTRt simulated by TruckSim. In addition, the value of the new rollover index and the LTRt reach 1 at the same time such that the RIt can also predict rollover risk of the triaxle bus in the Fishhook condition.

Figure 7
figure7

Rollover dynamics of the triaxle bus in Fishhook maneuver

Roll Stability of the Triaxle Bus

Roll Stability in Typical Maneuvers

To study the dynamic rollover stability of the triaxle bus, this section presents three typical rollover maneuvers, which include the J-turn maneuver, the Fishhook maneuver, and the Double-lane-change maneuver. The parameters of the triaxle bus are shown in Table 1.

J-turn maneuver. In this maneuver, the road adhesion coefficient is 0.85, and the front wheel steering angle is 6°.

As shown in Figure 8, the solid line, dashed line, and dotted line represent the roll dynamic response of the triaxle bus when the vehicle moves at 60 km/h, 80 km/h and 100 km/h, respectively. When the speed reaches 100 km/h, the value of the rollover index of the vehicle is over 1 at 2.1 s, and the vehicle is at risk of rollover. When the vehicle moves slowly, the value of RIt is also decreased. Therefore, when the steering angle input is a constant, with the higher speed, the vehicle is more likely to rollover.

Figure 8
figure8

RIt in J-turn maneuver

Fishhook maneuver. This is a severe rollover test condition. The amplitude of the front wheel steering angle is set to 4°, the road adhesion coefficient is 0.85.

In Figure 9, the triaxle bus will not be at risk of rollover at 60 km/h, 80 km/h and 100 km/h in the first phase of the Fishhook condition (before 3.5 s). In the second phase of the Fishhook condition, the bus rolls over at less than four seconds, with a speed of 100 km/h. Like the J-turn condition, the value of RIt decreases with the decrease of vehicle speed. Compared with the results of J-turn condition, the rollover happens easier in the Fishhook condition.

Figure 9
figure9

RIt in Fishhook maneuver

Double lane change (DLC) maneuver. The rollover stability in a DLC condition is also studied. The road adhesion coefficient is 0.85, and its width is 3.5 m. The movement track of triaxle bus in DLC condition is shown in Figure 10.

Figure 10
figure10

RIt in Double Lane Change maneuver

In DLC condition, although the value of RIt at 100 km/h is larger than the value of RIt at 80 km/h and 60 km/h, it will not roll over. Therefore, the triaxle bus has good roll stability in the DLC condition. Similar to the J-turn and Fishhook conditions, the value of RIt decreases with the decrease of vehicle speed, that is, reducing the vehicle speed is an effective measure to decrease the risk of the triaxle bus rollover in typical maneuvers.

In conclusion, the fishhook condition is the most dangerous. In the actual driving process, the driver should avoid repeatedly pounding the steering wheel. Furthermore, it is observed that the higher the vehicle speed, the worse the stability.

Factors of Vehicle Structure

To analyze the effect law of some key vehicle structure parameters on the roll stability, Figure 11 presents the stability regions with respect to the vehicle critical speed and a changeable vehicle structure parameter when other vehicle parameters are fixed as constants. The vehicle is asymptotically stable for the vehicle speed u, is less than the rollover critical speed uc, and unstable for u > uc.

Figure 11
figure11

Impact of various structure parameters on rollover stability: (a) Impact of distance between the second axle and the third axle; (b) Impact of ratio of the rear sprung mass to the unsprung mass; (c) Impact of angle between the roll axle and horizontal plane; (d) Impact of various torsion stiffness of vehicle body

As shown in Figure 11(a), the farther the distance between the second axle and the third axle, the smaller the stability region is. Thus, the shorter distance between them can be used to reduce rollover accidents in the future.

The sprung mass of vehicle will be heavier as the number of passengers increase. Figure 11(b) illustrates the impact of ratio of the rear sprung mass to the unsprung mass on rollover stability. Given 25 passengers, 60 kg per person and 7.5 kg of luggage per person in the rear space of triaxle bus, the ratio will increase about 0.9 when the triaxle bus is fully loaded and the critical speed will decrease about 9 km/h. Thus, when designing the triaxle bus, the sprung mass of the empty bus should be small enough to improve the rollover stability.

The roll axle is usually with a certain angle to the ground, the value of this angle also affects the roll performance. Figure 11(c) shows the impact of the angle between the roll axle and horizontal plane on rollover stability. It can be found that the rollover stability improves when the roll axle is low in the rear axle and high in the front axle.

The effects of torsion stiffness should be discussed to reflect the effect of roll coupling of each axle on rollover. As shown in Figure 11(d), larger torsion stiffness can reduce the difference between front rollover system and rear rollover system, thus enhance the rollover stability of triaxle bus.

In addition, this paper also does research on the rollover index under parameter uncertainties, especially tire parameters. Because equivalent sprung masses and equivalent wheel base is a function of cornering stiffness. These tire parameters can change as vertical tire forces vary. The vertical tire forces can change dramatically since the scenario dealt with in this paper is related to rollover and the mass of the bus can vary with a wide range depending on the number of passengers. So the impact of the cornering stiffness of tires on rollover stability is analyzed in Figure 12.

Figure 12
figure12

Impact of the cornering stiffness of tires on rollover stability: (a) Kf; (b) Km; (c) Kr

As shown in Figure 12(a), when the Kf increase, the rollover stability becomes worse. Because larger Kf can increase the yaw moment; To the contrary, the middle and rear wheels are located behind the center of mass. When the corning stiffness increases, the yaw moment decreases, thus the rollover stability is improved, as shown in Figure 12(b) and (c).

Factors of Vehicle Driving

The study of driving parameters is helpful to make suggestions on the driver’s operation. This section will analyze the influence of the driving parameters including the steering angle and road adhesion coefficient on rollover stability.

As shown in Figure 13(a), with the increase of steering angle, the maximum value of RIt is increased linearly, showing the critical value of steering angle is 6.8° when the vehicle speed is 60 km/h. Furthermore, the largest front wheel steering angle only changes 2.4°, but the RIt increases from 0.7 to 1. Therefore, reducing the largest front wheel steering angle has an obvious effect on improving the rollover stability.

Figure 13
figure13

Impact of different driving parameters on rollover stability. (a) Impact of front wheel steering angle; (b) Impact of road adhesion coefficient

Figure 13(b) represents the maximum value of RIt at various road adhesion coefficients. It shows that the maximum value of RIt grows almost linearly from 0.5 to 1.08 as the road adhesion coefficient increases before 0.625. When the road adhesion coefficient is too low, the generated lateral force is not enough to support the steering of the vehicle, resulting in less roll movement. When the road adhesion coefficient changes from 0.625 to 1.2, RIt exhibits a negative correlation with the road adhesion coefficient. Thus, when the road adhesion efficient is near 0.6, the rollover stability is poor.

Rollover Stability under External Disturbances

The triaxle bus encounters various interferences during driving, such as lateral force interference and vertical interference. Interference factors may lead the triaxle bus to roll over around the tipping point. Therefore, it is very important to understand the influence of interference factors on the rollover stability of the triaxle bus. The J-turn maneuver is selected to simulate rollover stability with external disturbance.

External Vertical Force

When the triaxle bus moves at 80 km/h, a pulse signal of vertical force is applied to a wheel from 2.0 s to 2.2 s. Figure 14(a) shows the maximum values of RIt when the external vertical force is applied to the inside wheel on different axles. The amplitude of the external vertical force on the front axle has little influence on vehicle rollover stability. If the vertical force is applied to the wheel of the second axle, the maximum value of RIt increases when the amplitude of the external vertical force increases. When it increases to 2700 N, the triaxle bus has rollover risk. The maximum value of RIt increases when the external vertical force on the third axle is less than 2300 N. It shows the external vertical force applied to the inside wheel will deteriorate the rollover stability, and when the disturbance is applied to the inside wheel on the third axle, the vehicle is much easier to rollover risk than when it is applied to the inside wheel on other axles.

Figure 14
figure14

Influence of vertical interference on rollover stability of the triaxle bus: (a) RIt with vertical force interference applied to the inside wheel; (b) RIt with vertical force interference applied to the outside wheel

Figure 14(b) shows the maximum values of RIt when the vertical force disturbs the outside wheel on different axles. With the increasing amplitude of the vertical force applied to the wheel on the second axle and the third axle, the RIt increases, meaning the rollover stability becomes worse. It also can be found that the vehicle is more likely to roll over when it is applied to the inside wheel on the third axle than when it is applied to the inside wheel on other axles. On the contrary, the larger amplitude of the vertical force applied to the wheel on the front axle, the vehicle will be more stable. Therefore, the vertical force applied to the outside wheel on the front axle can offset the rollover tendency.

Figure 15 shows the phase plane trajectory of the triaxle bus with the 6000 N vertical force on the outside wheel of the third axle. The trajectory shows that the vehicle is stable when the bus is not disturbed but the vehicle will roll over with the disturbance applied to the outside wheel on the third axle. It intuitively reflects that the vertical force applied to the wheels can affect the rollover stability.

Figure 15
figure15

Diagram of (\(\varphi - \dot{\varphi }\)) with vertical force interference

External Lateral Force

When the triaxle bus moves at 80 km/h, a pulse signal of lateral force is applied to a wheel from 2.0 s to 2.2 s. Figure 16(a) shows the maximum values of RIt when the lateral force interference is applied to the inside wheels on different axles. When the front axle is disturbed, the yaw rate of the triaxle bus will increase, causing over-steering and rollover. In Figure 16(a), the maximum value of RIt increases with the amplitude of the lateral force increasing. When the amplitude of the lateral force on the wheel of the second axle is less than 1800 N, or the amplitude of the lateral force on the wheel of the third axle is less than 2600 N, the greater the amplitude of the interference, the smaller the maximum value of RIt. Because the second axle and the third axle are located behind the center of gravity of the triaxle bus, the lateral interference will resist steering and then reduce the yaw rate of the triaxle bus. Thus, the rollover stability improves. If the lateral interference on the wheel of the second axle and the third axle continues to increase, the torque of lateral interference to the center of sprung mass is large enough to increase the rollover risk of the triaxle bus. Thus, the maximum value of RIt increases. In addition, the distance between the third axle and the center of gravity is further than that between the second axle and the center of gravity, therefore, the lateral interference to the third axle is more sensitive.

Figure 16
figure16

Influence of lateral interference on absolute maximum value of RIt without controlling: (a) RIt with lateral force interference applied to the inside wheel; (b) RIt with lateral force interference applied to the outside wheel

Contrary to Figure 16(a), when the second axle and the third axle are disturbed by the lateral force applied to the outside wheel, the yaw rate of the triaxle bus will increase and the triaxle bus has rollover risk. However, the lateral force on the front axle can reduce the over-steering to avoid the occurrence of rollover, as shown in Figure 16(b).

Figure 17 shows the phase plane trajectory of the triaxle bus subjected to 5000 N of the lateral force interference. When the triaxle bus is disturbed by the excessive lateral force on the outside wheel on the third axle, the roll angle and roll rate varies greatly.

Figure 17
figure17

Diagram of with (\(\varphi - \dot{\varphi }\)) lateral force interference

Conclusions

  1. (1)

    Taking the effect of torsion deformation of vehicle body into consideration, a new 6-DOF vehicle model was derived by dividing the triaxle bus into two parts, the second and third axle are equivalent to a virtual rear axle.

  2. (2)

    The equivalent distance from the center of gravity to the rear virtual axle is derived using the wheelbases of the second axle and the third axle. Furthermore, the torsion stiffness of the vehicle body is estimated.

  3. (3)

    A new rollover index was proposed to evaluate the rollover risk of the triaxle bus, and evaluated in some numerical cases, the results show that the proposed index has a good ability to detect rollover.

  4. (4)

    The stability analysis showed that the distance between the second axle and the third axle and the torsion stiffness of vehicle body have great influence on the rollover stability of the triaxle bus, and it is more prone to rollover when the lateral force interference applied to the wheel on the third axle.

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Authors’ Contributions

ZJ was in charge of the whole trial; JL and AK wrote the manuscript; YH assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thanks to Mr. Chao Wang of Nanjing University of Aeronautics and Astronautics for his critical discussion and reading during manuscript preparation.

Authors’ Information

Zhilin Jin, born in 1978, is currently an associate professor at Department of Vehicle Engineering, Nanjing University of Aeronautics and Astronautics, China. He received his PhD degree from Nanjing University of Aeronautics and Astronautics, China, in 2008. His research interests include vehicle system dynamics and control.

Jingxuan Li, born in 1997, is currently a master candidate at Department of Vehicle Engineering, Nanjing University of Aeronautics & Astronautics, China.

Yanjun Huang, born in 1986, is currently an assistant professor at Department of Mechanical & Mechatronics Engineering, University of Waterloo, Canada.

Amir Khajepour, born in 1962, is currently a professor at Department of Mechanical & Mechatronics Engineering, University of Waterloo, Canada.

Competing Interests

The authors declare no competing financial interests.

Funding

Supported by National Natural Science Foundation of China (Grant No. 51775269).

Author information

Correspondence to Zhilin Jin.

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Jin, Z., Li, J., Huang, Y. et al. Study on Rollover Index and Stability for a Triaxle Bus. Chin. J. Mech. Eng. 32, 64 (2019). https://doi.org/10.1186/s10033-019-0376-0

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Keywords

  • Rollover stability
  • Dynamic model
  • Triaxle bus
  • Rollover index
  • Vehicle active safety